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Teor. Veroyatnost. i Primenen., 2005, Volume 50, Issue 4, Pages 774–776 (Mi tvp131)  

This article is cited in 1 scientific paper (total in 1 paper)

Short Communications

On the convergence to uniform distribution

A. Ya. Kuznetsova

M. V. Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics

Abstract: This paper considers sums of independent identically distributed random variables. We give an example in which, under the unbounded growth of a number of summands, the probability densities $\tilde{p}_n(x)$ of fractional parts of these sums converge to 1 in the sense of
$$ \int_{0}^1|\tilde{p}_n(x)-1| dx\to 0, $$
but they do not converge to 1 in the uniform metric
$$ \sup_{0\leq x\leq 1}|\tilde{p}_n(x)-1|. $$


Keywords: fractional parts, random variables, uniform distributions, convergence “in variation”.

DOI: https://doi.org/10.4213/tvp131

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English version:
Theory of Probability and its Applications, 2006, 50:4, 687–689

Bibliographic databases:

Received: 15.11.2005

Citation: A. Ya. Kuznetsova, “On the convergence to uniform distribution”, Teor. Veroyatnost. i Primenen., 50:4 (2005), 774–776; Theory Probab. Appl., 50:4 (2006), 687–689

Citation in format AMSBIB
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\transl
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\yr 2006
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\pages 687--689
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    This publication is cited in the following articles:
    1. O. V. Viskov, V. I. Khokhlov, “Four areas of Yu. V. Prokhorov's studies and their perspectives”, Theory Probab. Appl., 60:2 (2016), 336–342  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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